LETTER Communicated by John Milton

Electrical Coupling Promotes Fidelity of Responsesin the Networks of Model Neurons

Georgi S. Medvedevmedvedev@drexel.eduDepartment of Mathematics, Drexel University, Philadelphia, PA 19104, U.S.A.

We consider an integrate-and-fire element subject to randomly perturbedsynaptic input and an electrically coupled ensemble of such elements.The latter is interpreted as either a model of electrically coupled popu-lation of neurons or a multicompartment model of a dendrite. Randomfluctuations blur the input signal and cause false responses in the sys-tem dynamics. For instance, under the influence of noise, the system mayrespond with an action potential to a subthreshold stimulus. We showthat the responses of the elements within the network are more reliablethan the responses of the same elements in isolation. Specifically, weshow that the variances of the stochastic processes generated by the cou-pled model can be made arbitrarily small (i.e., the network responses canbe made arbitrarily accurate) by increasing the number of elements in thenetwork and the strength of electrical coupling. Our results suggest thatthe organization of cells in electrically coupled groups on the networklevel, or the dendritic morphology on the cellular level, may be involvedin the filtering noise and therefore may play an important role in the in-formation processing mechanisms operating on the network or cellularlevel respectively.

1 Introduction

Neurons in many parts of nervous system interact by gap junctions (Bennett& Zukin, 2004; Connors & Long, 2004). Electrical coupling is also commonin physiological systems outside nervous system; for example, certain cellsin the heart and pancreas are connected by gap junctions (Keener & Sneyd,1998). Multicompartment models of individual neurons can also be viewedas electrically coupled networks (Segev & Burke, 1998; Dayan & Abbott,2001). Consequently, electrically coupled networks constitute a large classof models in mathematical biology. The goal of this letter is to point to a cer-tain general effect based on the interaction of electrical coupling and noise,provide a quantitative description of this phenomenon, and indicate rep-resentative applications. The main observation (principle) underlying thedynamical phenomena considered in this letter is that electrical couplingcan diminish the effects of noise on the neurons in the network. In some

Neural Computation 21, 3057–3078 (2009) C© 2009 Massachusetts Institute of Technology

3058 G. Medvedev

heuristic form, noise reduction is certainly intuitive. Electrical couplingacts as an averaging device: it distributes the noise between the coupled el-ements, making large fluctuations in an individual neuron less likely. In thisletter, we provide a quantitative description of this effect. Thus, when elec-trically coupled, neurons become much less sensitive to noise than when inisolation. Two applications of the reduction of noise by electrical couplingcan be found in the computational neuroscience literature. One, knownas the channel-sharing hypothesis, is used to account for the differencesin the firing patterns in isolated β−cells and those in electrically coupledislets of Langerhans due to the reduced effects of noise on the neuronsin the islets of Langerhans (Sherman, Rinzel, & Keizer, 1988; Sherman &Rinzel, 1991). The second example is a mechanism for the phasic episodesin the model of the locus coeruleus network in mammalian brain by mutualshunting of the uncorrelated noise in electrically coupled network (Usher,Cohen, Serven-Schreiber, Rajkowski, & Astor-Jones, 1999). We think thatthe scope of potential applications of the noise reduction by electrical cou-pling extends beyond these two examples and deserves a comprehensivemathematical study. It seems especially important to characterize the contri-butions of the principal network parameters such as network size, topology,and the strength of coupling to this important effect. In this letter, for net-works of electrically coupled integrate-and-fire (IF) neurons, we derive anestimate for the network variability in terms of network size, network topol-ogy, and strength of coupling. Our results show that under quite generalconditions and for a broad class of networks, the variability of the networkin the presence of noise can be made substantially smaller than that of asingle neuron under the same conditions. Moreover, in the limit of largenetwork size and strong coupling, the variability vanishes, thus making theneural responses practically independent from noise. These results suggestthat the organization of cells in electrically coupled groups on the networklevel, or the dendritic morphology on the cellular level, may be involvedin the filtering noise and therefore may play an important role in the infor-mation processing mechanisms operating on the network or cellular levelrespectively.

In section 2, we formulate an IF model of a single neuron and a modelof an electrically coupled network (ECN) of IF neurons. We deliberatelychose a very simple model for this study to make the analysis simpler andthe mechanism of the noise reduction transparent. The effects considered inthis letter can be easily reproduced using conductance-based models. How-ever, the analysis of the noise reduction in the ECNs of conductance-basedmodels of neurons has to deal with certain additional technical problems,which are not essential for the main phenomenon studied in this letter. Weplan to extend our results to ECNs of conductance-based models in thefuture. In section 3, we illustrate the implications of the reduced variabilityin the ECNs with two model problems. The first one is meant to demon-strate that the ability of the network to distinguish the signal from noise

Fidelity of Responses in the Networks of Neurons 3059

is much greater than that of a single IF neuron under the same conditionsand can be effectively controlled by varying network parameters such asthe size of the network and the strength of coupling. As the second exam-ple, we consider an electrically coupled population of spontaneously firingneurons and show that the rate of firing critically depends on the strengthof coupling. Each of these model problems is used to illustrate a generalmechanism, as it can be realized in many different modeling situations.Both examples in sections 3.1 and 3.2 rely on the reduced variability in theECNs and show that the latter can be an important factor shaping the net-work output. Thus, we conclude section 3 by formulating a general estimatecharacterizing the variability of the neural responses in ECNs in terms ofthe network parameters. This is the main result of this letter. Importantly,our estimate of the network variability explicitly shows the role of the net-work size, topology, and strength of coupling in reducing the variability ofneural responses. The estimate holds for a broad class of networks. In par-ticular, it yields effective estimates of neuronal variability in the networkswith nearest-neighbor and all-to-all coupling, two common types of net-work architecture. Section 4 contains the derivation of the main estimate.The final section reviews related work and discusses certain implicationsof the results of this letter.

2 The Model

Consider a nondimensional IF neuron model (Dayan & Abbott, 2001) sub-ject to weak white noise:

εvt = −vt + p(t) + √εσ wt, (2.1)

where vt is interpreted as a membrane potential, p(t) is an input signal, andwt is a standard Wiener process (Karatzas & Shreve, 1999). Small, positiveparameters ε and σ reflect the relative magnitudes of the membrane timeconstant and the noise intensity. Equation 2.1 describes the evolution ofthe membrane potential as long as it stays below threshold vth = 1. Uponreaching the threshold at time t = ta p−0, the system generates an actionpotential (AP),

vs = v+ > 1, s ∈ [ta p, ta p + �+),

followed by the refractory period,

vs = v− < 0, s ∈ [ta p + �+, ta p + �+ + �−].

After that, the evolution of the system is again governed by equation 2.1until the next AP. Positive parameters �+ and �− denote the durations of

3060 G. Medvedev

the AP and the refractory period, respectively. The input signal p(t) will bespecified below.

Next, we introduce the ECN of IF neurons,

εv(i)t = −v

(i)t + I (i)

c + p(t) + √εσ w

(i)t , i = 1, 2, . . . N, (2.2)

where I (i)c stands for the current that cell i receives from other cells:

I (i)c =

N∑j=1

gi j(v

( j)t − v

(i)t

). (2.3)

The nonnegative coupling weight gi j corresponds to the conductances ofthe gap junction between cell i and cell j . In the vector form, equation 2.2can be rewritten as

εVt = −Vt + P(t) + gDVt + √εσ Wt. (2.4)

Here, Vt = (v(1)t , v

(2)t , . . . , v

(N)t )T , P(t) = (p(t), p(t), . . . , p(t))T , Wt = (w(1)

t ,

w(2)t , . . . , w

(N)t )T , and w

(i)t are independent copies of the standard Wiener

process. We refer to parameter

g = maxi, j

gi j (2.5)

as the strength of coupling. The matrix

D = (di j

)Ni, j=1 , di j =

{g−1gi j , i �= j,

−g−1 ∑i �= j gi j , i = j.

reflects the connectivity of the network. The latter is important. To illustratethe role of the network topology in shaping the network output, we willuse the following examples.

Example 1. The nearest-neighbor coupling is a representative example ofthe local connectivity. In this network configuration, each cell in the interiorof the array is coupled to two nearest neighbors:

I ( j)c = (

v( j+1)t − v

( j)t

) + (v

( j−1)t − v

( j)t

), j = 2, 3, . . . , N − 1.

The coupling currents for the cells on the boundary are given by

I (1)c = v

(2)t − v

(1)t and I (N)

c = v(N−1)t − v

(N)t .

Fidelity of Responses in the Networks of Neurons 3061

This yields the following coupling matrix:

D =

⎛⎜⎜⎜⎝

−1 1 0 . . . 0 0

1 −2 1 . . . 0 0

. . . . . . . . . . . . . . . . . .

0 0 0 . . . 1 −1

⎞⎟⎟⎟⎠ . (2.6)

Example 2. The all-to-all coupling features global connectivity:

I ( j)c =

N∑i=1

(v

(i)t − v

( j)t

), j = 1, 2, 3, . . . , N. (2.7)

Thus,

D =

⎛⎜⎜⎜⎝

−N 1 1 . . . 1 1

1 −N 1 . . . 1 1

. . . . . . . . . . . . . . . . . .

1 1 1 . . . 1 −N

⎞⎟⎟⎟⎠ . (2.8)

We emphasize that our results apply to a broad class of networks. Thenetworks in examples 1 and 2 are used only to illustrate the theory.

Remark 1. Rescaling the leaky IF model leading to equation 2.1 is straight-forward (Chow & Kopell, 2000; Lewis & Rinzel, 2003; Gao & Holmes, 2007).We thus omitted the details. We deliberately chose a very simple dynamicalequation, 2.1, for our model to make the proposed mechanism transparentand certain calculations explicit. Equation 2.1 approximates the subthresh-old dynamics of a broad class of conductance based models. Moreover,neither the linearity of the right-hand side in v nor the smallness of ε > 0 isessential for the proposed mechanism. These features are kept in the modelfor analytical convenience.

Remark 2. An alternative interpretation of the coupled model with thenearest-neighbor coupling, equation 2.6, is to view each variable v

(i)t , i =

1, N as the membrane potential of an equipotential compartment of a spa-tially extended model of a dendrite. Then the coupling term representsthe longitudinal currents between adjacent compartments (Segev & Burke,1998; Dayan & Abbott, 2001). In certain neurons, under strong synapticinput, an AP can be generated on active dendrites and then propagateto the soma (Spruston, Stuart, & Hausser, 1999; Hanson, Smith, & Jaeger,2004). Thus, equation 2.4 can be viewed as a model of spike initiation and

3062 G. Medvedev

propagation from a dendritic location or spontaneous subthreshold activity(Fatt & Katz, 1950).

3 Reduction of Noise by Electrical Coupling

In section 4, we will prove a very general and important property of ECNs:the noise reduction principle. It shows that in electrically coupled groups,neurons are less affected by noise than when in isolation. The magnitude ofthis effect depends on network size and topology and the coupling strength.The analysis in section 4 characterizes the contributions of these parametersto the network output. We consider two implications of noise reduction forthe network dynamics: one for the information processing in ECNs andanother for pattern formation. In this section, we formulate two modelproblems designed to highlight both aspects of noise reduction in ECNs.

3.1 The Fidelity Problem. Assume that the input signal consists of afinite sequence of square pulses of duration � � ε and amplitude 1 ± δ,

0 < δ < 1 delivered at times ti > 0, i = 1, m:

p(t) =m∑

i=1

(1 + κiδ)1[ti ,ti +�](t), κi = ±1. (3.1)

Here, ti+1 − ti ≥ 2T > 0, i = 1, m and � < T for some T > 0, and 1A standsfor the characteristic function of A ⊂ R. If κi = 1, pulse i has amplitude1 + δ > 1. We call such a pulse strong because it elicits an AP when appliedto an IF neuron in the rest state without noise (σ = 0). In contrast, a weakpulse of amplitude 1 − δ fails to evoke an AP under the same conditionsprovided ε > 0 is sufficiently small. If ε > 0 is small, the deterministic sys-tem recovers from receiving a pulse in time O(ε), and therefore it respondsto each strong pulse in the train, equation 3.1, with an AP and ignores weakones (see Figure 1a). The presence of noise introduces the probability offalse responses. For instance, the system may fire an AP in response to aweak stimulus (see Figure 1b). By the fidelity of the randomly perturbedsystem (σ > 0), we mean its ability to differentiate between strong andweak pulses. The numerical examples in Figure 1 show that electrical cou-pling enhances the fidelity of neural responses. Plots in Figures 1b and 1cshow typical responses of an uncoupled population to three weak pulses.The noise intensity σ = 0.1 used in this example is sufficient to impair thesystem’s ability to differentiate between weak and strong pulses. The plotin Figure 1b shows the time series of a randomly chosen cell from the un-coupled population: the cell fires APs to all three weak pulses. The plot inFigure 1c shows the population response. The responses of the populationcoupled with nearest-neighbor coupling and g = 10 are markedly differentfrom those of the uncoupled population. The population responds to only

Fidelity of Responses in the Networks of Neurons 3063

one out of three pulses (see Figures 1d and 1e). The fact that all neurons firein response to the second pulse is due to the synchronization imposed bythe electrical coupling: once one neuron fires an AP, it triggers all remainingneurons to fire (see Figure 1e). The comparison of the time series generatedby cells from the uncoupled and coupled populations shows that the lat-ter fluctuate much less than the former (compare Figures 1b and 1d). Thevariability of the time series becomes smaller for larger g. The voltage tracesampled from the population with g = 25 follows the input signal tightly(see Figure 1f). This trend continues for larger values of g (see Figure 1g).At these levels of coupling strength, the network does not make a singlefalse response. For larger g, the behavior of the cells in the network becomesclose to the deterministic IF model, 2.1, with σ = 0. Thus, in the presence ofnoise, the electrical coupling reduces the variability of the neural responses,thereby making them more predictable.

3.2 Spontaneous Firing. The population firing rate is a basic measureof activity in neural networks. It plays an important role in both theoreticaland experimental studies because the variations in the firing rate in neuralnetworks often signal changes in the physiological or cognitive state of theanimal. For example, the rate of irregular firing of the dopamine neuronsin mammalian midbrain correlates with the rate of the dopamine release(Grace & Bunney, 1984), and the rate of firing of the neurons in the locuscoeruleus network in the mammalian brain stem is correlated with the rateof the norepinephrine release (Berridge & Waterhouse, 2003). Moreover, thevariations in the rates of firing in the dopamine neurons and in the locuscoeruleus network code for the prediction of reward (Schultz, Dayan, &Montague, 1997) and mark the transition to a more alert state (Aston-Jones,Rajkowski, & Cohen, 2000; Berridge & Waterhouse, 2003), respectively. Therates of irregular firing in dopamine and norepinephrine neurons are justtwo representative examples of the coding of the physiological and cog-nitive states by the rate of the neural activity. Characterizing dynamicalmechanisms controlling the firing rate in neural populations is an importantproblem in theoretical neuroscience. Below, we consider a model problemthat elucidates the key factors controlling the rate of spontaneous firing inECNs. Specifically, we consider a population of IF neurons coupled elec-trically in the presence of noise, equation 2.3, with P(t) ≡ 0. If no noise ispresent (σ = 0), the coupled system remains silent. In the presence of noise,the neurons become spontaneously active. Clearly the rate of firing de-pends on the noise intensity: the cells are more likely to fire under strongerstochastic forcing. Importantly, the firing rate critically depends on such net-work attributes as size, topology, and coupling strength. Plots in Figure 2show that the firing rate can change dramatically under moderate changein the strength of coupling. Thus, the modulation of the coupling strengthpresents an interesting mechanism for frequency control. This mechanism

3064 G. Medvedev

Fidelity of Responses in the Networks of Neurons 3065

Figure 2: The strength of coupling can effectively control the rate of sponta-neous firing in ECN, equation 2.3, in the the presence of noise. (a–c) Plots showspontaneously active ECNs for different levels of the coupling strength. Notethat the rate slows dramatically under moderate changes in the strength ofcoupling (d). The parameter values used for this figure are the same as shownin the caption to Figure 1, except σ = 1.0.

Figure 1 (opposite): The fidelity of neural responses (see section 3.1). (a) Thedeterministic model, 2.1, fires in response to the strong pulse and ignores theweak ones. (b) In the presence of small noise, the system can fire in response tothe weak pulse. All three pulses shown in b are weak. The APs are due to therandom fluctuations. (c) The response of the uncoupled population to the threeweak pulses when the noise intensity is σ = 0.1. The noise intensity remains thesame in the experiments shown in (b–g). (d, e) The electrical coupling markedlyreduces the fluctuations due to the noise and thus enhances the fidelity of theneural responses. The neurons are coupled via nearest-neighbor coupling. Thecoupling strength used in these experiments is g = 10. (f, g) The fluctuations aresmaller for stronger coupling. For g = 25, the coupling practically annihilatedthe effect of noise on the system dynamics. Compare the fluctuations in b andf . The values of the other parameters used for this figure: ε = 0.2, �+ = 0.2,�− = 0.8.

3066 G. Medvedev

was suggested to be responsible for the switches between phasic and tonicfiring in the locus coeruleus network (Usher et al., 1999).

3.3 The Noise Reduction Principle. The examples in sections 3.1 and3.2 clearly show that reduced variability in ECNs can be an importantfactor shaping the network output. In this section, we formulate a generalestimate characterizing the variability of the neural responses in terms ofthe network parameters. Our estimate of the network variability explicitlyshows how the response properties of the network (see section 3.1) and thefiring patterns (see section 3.2) depend on the network size and topology,and the strength of coupling.

To highlight the main ingredients of the mechanism of the noise reduc-tion, we consider a slightly more general model than equation 2.4:

εVt = −Vt + D(g)Vt + P(t) + √εσ Wt. (3.2)

Here, we preserve the notation used in equation 2.4. For the N × N couplingmatrix D(g) depending on the parameter g ≥ 0, we assume:

Condition A:

ker D(g) = Span {e}, e = (1, 1, . . . 1)T . (3.3)

Condition A is simply Kirchoff’s law. To formulate our second condition,we need the following auxiliary (N − 1) × N matrix:

S =

⎛⎜⎜⎜⎝

−1 1 0 . . . . . . 0 0

0 −1 1 . . . . . . 0 0

. . . . . . . . . . . . . . . . . . . . .

0 0 0 . . . . . . −1 1

⎞⎟⎟⎟⎠ . (3.4)

For D(g) satisfying condition A, one can find an (N − 1) × (N − 1) matrixK (g) such that

SD(g) = K (g)S (3.5)

(see the appendix). Denote −ν(g) the largest eigenvalue of K (g), where

K (g) = 12

(K (g) + K T (g)). (3.6)

As our second assumption on the coupling matrix D(g), we require that

Condition B:

limg→∞ ν(g) = ∞. (3.7)

Fidelity of Responses in the Networks of Neurons 3067

This condition means that the coupling is dissipative for large g and thatthe dissipation rate can be controlled by g. As will be shown below, thiscondition is also natural for ECNs.

Under these conditions, ECN (see equation 3.2) satisfies the followingnoise-reduction principle.

Theorem 1. Let Vt, t ≥ 0, be a solution of equation 3.2 such that E (V0 −E V0)(V0 − E V0)T is finite. Suppose conditions A and B hold. Then for g suf-ficiently large,

limt→∞

max1≤k≤N

var v(k)t ≤ σ 2

(1N

+ κ(g, N)(1 + og(1)

)),

κ(g, N) = −tr {K −1(g)}N, (3.8)

where K (g) is defined in equation 3.5 and = SST . In particular, for g � 1, κ(g)is positive and

limg→∞

κ(g, N) = 0. (3.9)

Remark 3. The statement of the theorem is true for any finite ε > 0. How-ever, when ε > 0 is small, the variances of v

(k)t approach their asymptotic

values very fast. In particular, one can rewrite equation 3.8 as follows:

max1≤k≤N

var v(k)t ≤ σ 2

(1N

+ κ(g, N)(1 + og(1)

) + O(ε))

,

for t ≥ O(−ε ln ε). (3.10)

Remark 4. For readers’ convenience, we present the explicit expression of(N − 1) × (N − 1) matrix:

:= SST =

⎛⎜⎜⎜⎝

2 −1 0 . . . 0 0

−1 2 −1 . . . 0 0

. . . . . . . . . . . . . . . . . .

0 0 0 . . . −1 2

⎞⎟⎟⎟⎠ . (3.11)

Note that − can be interpreted as the discrete Laplacian. The eigenvaluesof are given by

ωk = 4 sin2 kπ

2N, k = 1, 2, . . . , N − 1. (3.12)

3068 G. Medvedev

In particular, for the trace of , we have

tr =N−1∑k=1

ωk = π N + O(1). (3.13)

Corollary 1. For the nearest-neighbor coupling (see example 1),

κ(g, N) = g−1 N2. (3.14)

Corollary 2. For the all-to-all coupling (see example 2),

κ(g, N) = g−1tr = πg−1 N + O(1). (3.15)

Estimate 3.8 is the key to understanding the role of electrical coupling incounteracting the effects of noise. The first term on the right-hand side ofequation 3.8 can be made arbitrarily small by increasing N, while the secondterm can be made small by increasing the strength of coupling g. Therefore,the combination of strong coupling and sufficiently large size of the networkcan weaken the effect of noise on the network performance to an arbitrarilysmall degree. Moreover, equation 3.8 captures the topology of the networkthrough κ(g, N), which reflects the density of connections in the network.The magnitude of gκ(g, N) can vary from O(N) for local nearest-neighborcoupling to O(1) for global all-to-all coupling (see corollaries 1 and 2).

4 The Analysis

In this section, we derive estimate 3.8 characterizing the variability of thecoupled system. The derivation proceeds in three steps:

Step 1. We analyze the system of equations for the differences η(i)t =

v(i+1)t − v

(i)t , i = 1, N − 1. It is obtained by subtracting equation

i from equation (i + 1) in the system of equations 3.2,

ηt = Aηt + σ√ε

SWt A = ε−1 (K (g) − I ), (4.1)

where (N − 1) × (N − 1) matrix K (g) satisfies equation 3.5, Wt

stands for the N-dimensional Brownian motion as before, I is an(N − 1) × (N − 1) identity matrix, and ηt = (η(1)

t , η(2)t , . . . , η

(N−1)t ).

Step 2. We consider the averaged equation,

εξt = −ξt + p(t) + σ√

ε

NXt, Xt =

N∑i=1

w(i)t , (4.2)

where ξt = N−1 ∑Ni=1 v

(i)t .

Fidelity of Responses in the Networks of Neurons 3069

Step 3. We combine the results for equations 4.1 and 4.2 to obtain equa-tion 3.8.

In the remainder of this section, we implement these steps.

4.1 Step 1.

A := 12

(A+ AT ). (4.3)

Denote the largest eigenvalue of A by

λ := −ε−1μ, (4.4)

where μ = 1 + ν(g) and −ν(g) is the largest eigenvalue of K (g) (cf.equation 3.6). By equation 3.7, for large g,

λ ≈ −ε−1ν(g) < 0. (4.5)

Since A in equation 4.1 is symmetric, so is e2t A. Thus,

|e2t A| = e−μt

ε , t ≥ 0. (4.6)

Here, |·| stands for the operator matrix norm induced by a Euclidean vectornorm in R

N−1 (Horn & Johnson, 1985). The solution of equation 4.1 is agaussian process,

ηt = et Aη0 + σ√ε

∫ t

0e (t−u)ASdWu, (4.7)

whose mean vector M(t) = E ηt and covariance matrix V(t) = E (ηt −M(t))(ηt − M(t))T have the following representations (Karatzas & Shreve,1999):

M(t) = et AM(0), (4.8)

V(t) = et A(

V(0) + σ 2

ε

∫ t

0e−uAe−uAT

du)

et AT, (4.9)

3070 G. Medvedev

where is defined in equation 3.11. Next, we estimate the trace of V(t),tr {V(t)}. For this, we note that

tr{∫ t

0e (t−u)Ae (t−u)AT

du}

=∫ t

0tr {euAeuAT }du =

∫ t

0tr {e2uA}du = tr

{

∫ t

0e2uAdu

}

= 12

tr {A−1(e2t A − I )}. (4.10)

By plugging equation 4.10 in 4.9 and by taking into account equation 4.6,we have

limt→∞ tr {V(t)} = −σ 2

2εtr {A−1}. (4.11)

Furthermore, from the definition of A in equations 4.1 and 4.3, and assump-tion 3.7, for g � 1, the expression on the right-hand side of equation 4.11can be rewritten as

limt→∞ tr {V(t)}= 1

2σ 2 N−1κ(g, N)

(1 + og(1)

),

(4.12)κ(g, N) = −tr {K −1(g)}N,

where og(1) denotes terms, which vanish as g → ∞.

4.2 Step 2. We turn to equation 4.2. By noting that Xt ∼ √Nwt , from

equation 4.2, we have

εξt = −ξt + p(t) + σc√

εwt, σc = σ√N

. (4.13)

Thus,

var ξt = σ 2

Nε

∫ t

0e

2(s−t)ε ds = σ 2

2N

(1 − e

−2tε

)→ σ 2

2N, t → ∞. (4.14)

4.3 Step 3. The following relations are derived from the definitions ofξt and η

(i)t :

v(N)t = ξt + N−1

N∑i=1

iη(i)t , (4.15)

v(k)t = v

(k+1)t − η

(k)t , k = N − 1, N − 2, . . . , 1. (4.16)

Fidelity of Responses in the Networks of Neurons 3071

Equation 4.15 implies

var v(N)t = var ξt +

N−1∑i=1

2icov (ξt, η(i)t )

N+

N−1∑i, j=1

i j N−2cov(η

(i)t , η

( j)t

).

(4.17)

The terms on the right-hand side of equation 4.17 comply with the followingbounds:

N−1N−1∑i=1

2icov(ξt, η

(i)t

) ≤ N−1N−1∑i=1

(var ξt + i2var η

(i)t

)≤ var ξt + Ntr {V(t)}, (4.18)

N−1∑i, j=1

i j N−2cov(η

(i)t , η

( j)t

) ≤N−1∑i, j=1

var η(i)t ≤ Ntr {V(t)}. (4.19)

The combination of equations 4.16 to 4.19 yields

var v(N)t ≤ 2 (var ξt + N tr {V(t)}) . (4.20)

The estimates for var v(k)t , k = 1, N − 1 are derived similarly. The combina-

tion of equations 4.15 and 4.16 yields

v(N−k)t = ξt + N−1

N−k−1∑i=1

iη(i)t + N−1

N∑i=N−k

(i − N)η(i)t , k = 1, N − 1,

(4.21)

By following the steps, which we used to arrive at equation 4.20, we obtainfrom equation 4.21,

max1≤k≤N

var v(k)t ≤ 2 (var ξt + N tr {V(t)}) . (4.22)

After plugging equations 4.12 and 4.14 in 4.22, we obtain equation 3.8.

Finally, the modification of equation 3.8 stated in remark 3, follows fromequations 4.6 and 4.14. Corollaries 1 and 2 follow after noting that for thenearest-neighbor and all-to-all coupling, K (g) are equal to −g and −gNI ,respectively.

3072 G. Medvedev

5 Discussion

Models of single neural cells and networks feature a rich variety of pat-terns of electrical activity. Often neuronal models are located close to thetransitions between different stable regimes, where the output of the model(e.g., the frequency of oscillations for a single model or the mean firing ratefor the population) is very sensitive to small perturbations. Under these con-ditions, even small noise becomes an important factor in pattern formation.The manifest role of noise in shaping activity patterns is well recognizedin theoretical and experimental neuroscience (Fatt & Katz, 1950; Verveen& DeFelice, 1974; Knight, 1972a, 1972b; Chow & White, 1996; Fox, 1997;White, Rubenstein, & Kay, 2000; Hitczenko & Medvedev, 2009). There aretheoretical studies suggesting possible synergistic roles for noise in shapingneuronal responses. For example, the responses of neural systems to inputsignals of certain types can be optimized in the presence of noise at certainlevels of intensity via stochastic resonance type mechanisms (Wiesenfeld &Moss, 1995; Longtin & Hinzer, 1996; Collins, Chow, & Imhoff, 1995; Longtin,1997), stochastic input can synchronize or desynchronize neural activity(Goldobin & Pikovsky, 2005a, 2005b, 2006; Ermentrout, Gallan, & Urban,2008; Danzl, Hansen, Bonnet, & Moehlis, 2008). Importantly, fluctuatingstimuli evoke reliable and reproducible responses in neocortical neurons(Mainen & Sejnowski, 1995) and unveil coexisting spike patterns in neu-ronal responses (Fellous, Tiesinga, Thomas, & Sejnowski, 2004). On theother hand, there is a line of research revealing the mechanisms by whichneural networks composed of individual neurons having very modest in-formation processing qualities and subject to noise can nonetheless achieveprecise overall performance. For example, it was shown that inhibitory cou-pling can improve the system dynamic range and the signal-to-noise ratio inthe networks of IF neurons (Mar, Chow, Gerstner, Adams, & Collins, 1999).Similarly, the interspike interval correlations can decrease the noise powerat low frequencies and improve information transfer (Chacron, Lindner, &Longtin, 2004). The mechanism of the noise-reduction in ECNs, presentedin this letter, offers another way by which neural networks can counteractthe effects of noise. Typically the variability of responses in neuronal net-works exceeds that of an individual neuron. It is a remarkable propertyof ECNs to provide a mechanism for overall reduction of noise in the net-work, which can result in the network variability being significantly lowerthan the variability of a single neuron under the same conditions. Specif-ically, uncorrelated noise acting on individual neurons can be effectivelycontrolled by the network parameters such as the strength of coupling andthe network size. Moreover, our analysis explicitly accounts for the con-tribution of the network topology to the reduction of noise by electricalcoupling. We complemented the analysis of ECNs with the numerical re-sults for two model problems designed to illustrate the potential roles ofthe noise reduction by electrical coupling in shaping neural responses. The

Fidelity of Responses in the Networks of Neurons 3073

fidelity problem, considered in section 3.1, suggests how electrical couplingcan be used to enhance the information processing properties of the system,while the spontaneous firing example is representative for the effects of themodulation of the strength of coupling on the firing patterns generated bythe network in the presence of noise.

The dynamical effects of the electrical coupling in the context of thepattern formation in neuronal networks have been studied using severaldistinct sets of techniques: the theory for weakly connected networks(Ermentrout & Kleinfeld, 2001; Kopell & Ermentrout, 2002, 2004; Lewis &Rinzel, 2003; Pfeuty, Mato, Golomb, & Hansel, 2003; Galan, Ermentrout,& Urban, 2005; Mancilla, Lewis, Pinto, Rinzel, & Connors, 2007), theanalysis of the Poincare map (Chow & Kopell, 2000; Lewis & Rinzel, 2003;Medvedev & Cisternas, 2004; Gao & Holmes, 2007), and constructinga Lyapunov function (Medvedev & Kopell, 2001; Medvedev, Wilson,Callaway, & Kopell, 2003). In this study, we considered the case of strongelectrical coupling, which typically results in synchronization of activityacross the network. Synchronization mediated by electrical couplingcontributes to a range of important physiological and cognitive functionssuch as γ−oscillations, which are thought to be important for informationprocessing in the brain (Traub, Kopell, et al., 2001), synchronizationof inhibition in neocortex (Beierlein, Gibson, & Connors, 2000; Hestrin& Galarreta, 2005), the mechanism of the control of sleep-wake cycle(Garcia-Rill, Heister, Ye, Charlesworth, & Hayar, 2007), and the mechanismof attention (Aston-Jones et al., 2000), as well as for certain pathologies, suchas the onset of seizures (Traub, Whittington, et al., 2001). The mechanismof noise reduction analyzed in this letter is closely related to the problemof synchronization. Specifically, equation 4.8 implies that

limt→∞ m(i)

t = 0, m(i)t = E

(v

(i+1)t − v

(i)t

), i = 1, 2, . . . , N − 1. (5.1)

The reasons by which equation 5.1 holds is exactly the same as is oftenused to prove synchronization in diffusively coupled sets of deterministicdifferential equations. Namely, all eigenvalues of the matrix of the systemof equations for η

(i)t = v

(i+1)t − v

(i)t , i = 1, 2, . . . , N − 1, A (see equation 4.1)

have negative real parts after possibly taking g sufficiently large. In fact,our conditions A and B are natural conditions for synchronization indeterministic ECNs. The estimate of tr V(t) in equation 4.12 can be usedto show synchronization in the networks of stochastically forced networksof IF neurons (and in much more general class of problems). Indeed, fromequation 4.12 via Chebyshev inequality, we have

P{∣∣v(i+1)

t − v(i)t − m(i)

t

∣∣ > α} ≤ σ 2κ(g, N)(1 + og(1))

2α2 → 0, as g → ∞,

(5.2)

3074 G. Medvedev

for any α > 0 and t > 0. The combination of equations 5.1 and 5.2 can beviewed as a form of synchronization for the population of stochasticallyforced elements.

ECN (see equation 2.4) with nearest-neighbor coupling (see equation 2.6)can be interpreted as a compartmental model of a spatially extended den-drite subject to electrical noise. It is easy to generalize this model to ac-count for more complex dendritic geometries (e.g., to incorporate brancheddendrites). The magnitude of the electrical noise depends on cell size. Inparticular, it more strongly affects the finer parts of the dendrite (Fatt &Katz, 1950) and may interfere with the integration of synaptic input atdistal dendritic locations. The noise reduction principle implies that theeffects of noise can be significantly diminished by electrical coupling be-tween dendritic compartments. This suggests that dendritic morphologymay play an important role in the mechanism for filtering electrical noiseand increasing the precision of synaptic integration, and dendritic spikeinitiation (Spruston et al., 1999). This is complementary to the recognizedroles of the dendritic morphology in integrating synaptic inputs (Shepherd& Koch, 1998) and in shaping the firing patterns (Pinsky & Rinzel, 1994;Mainen & Sejnowski, 1996).

Appendix

Lemma 1. Let D be an (N × N) matrix and S is as in equation 3.4. Suppose that

De = 0, e = (1, 1, . . . , 1)T . (A.1)

Then there exists an (N − 1) × (N − 1) matrix K such that

SD = K S. (A.2)

If, in addition,

ker (D) = Span {e}, (A.3)

then K is invertible.

Proof. Recall that SST is invertible (see remark 4) and define

K = SDST (SST )−1. (A.4)

Below we show that matrix K defined by equation A.4 satisfies the proper-ties stated in the lemma.

Fidelity of Responses in the Networks of Neurons 3075

From equation A.4, we have

K SST = SDST (A.5)

and

K S(ST S) = SD(ST S). (A.6)

For a given x ∈ RN, let x′ ∈ R(ST S) and x′′ ∈ ker (ST S) be such that x =

x′ + x′′. Note that

Sx′′ = 0, (A.7)

because ker (S) = ker (ST S). From equations A.6 and A.7, we have

K Sx = K S(x′ + x′′) = K Sx′ = SDx′ = SDx ∀x ∈ RN.

This shows equation A.2.Finally, if equation A.3 holds, then

rank (SD) = N − 1.

This and equation A.2 imply that rank (K ) = N − 1, that is, K is invertible.

Acknowledgments

Thanks to Dmitry Kaliuzhnyi-Verbovetskyi for helpful discussions, KarenMoxon and Andrey Olypher for reading an earlier version of the manuscriptand useful comments, and the anonymous referees for numerous sugges-tions, which helped to improve the letter. The preliminary results presentedin this letter were reported at the Seventeenth Annual Computational Neu-roscience Meeting, CNS*2008 (Medvedev, 2008).

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Received July 8, 2008; accepted April 7, 2009.